LETTER TO THE EDITOR: Transition temperature of the weakly interacting Bose gas: perturbative solution of the crossover equations in the canonical ensemble
We compute the shift of the critical temperature Tc with respect to the ideal case for a weakly interacting uniform Bose gas. We work in the framework of the canonical ensemble, extending the criterion of condensation provided by the canonical particle counting statistics for the zero-momentum state of the uniform ideal gas. The perturbative solution of the crossover equation to lowest order in powers of the scattering length yields (Tc-Tc0)/Tc0 = -0.93aρ1/3, where Tc0 is the transition temperature of the corresponding ideal Bose gas, a is the scattering length, and ρ is the particle number density. This result is at variance with the standard grand canonical prediction of a null shift of the critical temperature in the lowest perturbative order. The non-equivalence of statistical ensembles for the ideal Bose gas is thus confirmed (at the lowest perturbative level) also in the presence of interactions.